This is a clinical trial for subjects with hematologic malignancies ( acute myelogenous leukemia, acute lymphocytic leukemia, Hodgkin's or Non-Hodgkin's lymphoma, or myelodysplastic syndrome) who are in need of a donor stem cell transplant, and for whom an umbilical cord blood transplant is thought to be the best option. For allogeneic transplant donors, we typically try to use related family members, such as brothers or sisters, or volunteer donors who are 'HLA matched', i.e. share similar proteins on their cells. This study is for subjects for whom such a matched sibling donor or a matched unrelated donor is not available.